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1 One-Sided Commitment in Dynamic Insurance Contracts: Evidence from Private Health Insurance in Germany October 18, 2011 Annette Hofmann Mark Browne University of Hamburg University of Wisconsin-Madison Institute for Risk and Insurance Gerald D. Stephens CPCU Chair Faculty of Economics and Social Sciences in Risk Management and Insurance Business Administration Department Wisconsin School of Business Von-Melle-Park University Avenue Hamburg (Germany) Madison, WI (USA) Abstract This paper studies long-term private health insurance (phi) in Germany. It describes the main actuarial principles of premium calculation and relates these to existing theory. In the German phi policyholders do not commit to renewing their insurance contracts, but insurers commit to oering renewal at a premium rate that does not reect revealed future information about the insured risk. We show that empirical results are consistent with theoretical predictions: front-loading in premiums generates a lock-in of consumers, and more front-loading is generally associated with lower lapsation. Due to a lack of consumer commitment, dynamic information revelation about risk type implies that high-risk policyholders are more likely to retain their phi contracts than are low-risk types. JEL Classication: G22 I11 I18 Keywords: private health insurance; one-sided commitment; guaranteed renewable insurance.

2 1 Introduction In many European countries, health insurance is oered by government-sponsored social insurance funds. However, these funds are subject to mounting problems of nancing the welfare state, and do not oer much, if any, consumer choice. Leading policy-makers look for alternative strategies for creating incentives to increase eciency. In some countries, such as the United States, health insurance is mostly oered by private companies and premiums are riskbased. As rst discussed by Arrow (1963), in this environment, individuals face premium or reclassication risk. 1 Premium risk is the risk of an increase in the health insurance premium when the policyholder's health deteriorates. Given that in the United States, a large part of the population is insured via private health insurance, premium risk is a major problem since long-term health insurance is not available at a guaranteed price. 2 This situation shows that freely competitive health insurance markets do not seem to work eciently either. In these markets, policy-makers are often concerned about selection, equity, and access issues. Thus, creating an environment that can overcome current ineciencies in health insurance markets appears to be a challenging task. The present paper addresses the issue of long-term insurance contracts in a world where risk types evolve over time. This issue is especially important in the health and life insurance industries. The rst best insurance policy in this world would fully insure against both the short-term risk of an accident or illness and the long-term reclassication risk (premium risk). In a private health insurance market, it is far from easy to design long-term contracts that protect consumers from premium risk in the long run. This reclassication risk mainly emerges because contracts tend to be incomplete, i.e., they do not specify a price for every possible state of the world. To date, the insurance economics literature oers two solutions to this problem. On the one hand, Pauly et al. (1995) propose solving the premium risk problem via guaranteed renewable insurance contracts. Since policyholders initially prepay guaranteed renewable premiums to cover losses of everyone in the pool who (will) become high risk, the leaving of a low-risk policyholder has no impact on the insurer's prots while the leaving of a high-risk policyholder is protable to the insurer. Yet the technical design of long-term guaranteed renewable contracts seems sophisticated. On the other hand, Cochrane (1995) argues that the premium risk problem could be solved via separate premium insurance that pays an indemnity in the event an individual becomes a high risk. This paper looks at the German private health insurance market, a regulated market with guaranteed renewable premiums. It summarizes the main actuarial principles of premium calculation and discusses some of the major issues. The objective of the paper is threefold. First, it describes the German private health insurance experience (insurance contracts, actuarial premium calculation model, regulation, and market structure). Second, it relates this experience to existing theory (it shows how the oered contracts and the way they are priced relates to theoretical predictions). Third, it provides evidence on the relevance of the theory through individual data. In particular, we aim to investigate whether one-sided commitment contracts can solve well-known issues in health insurance. Using individual-level data, our study contributes to the literature on dynamic contract theory in two ways. First, it conrms theoretical ndings on dynamic contracts and one-sided commitment. Second, private health insurance in Germany is signicantly dierent from that in other countries which makes Ger- 1 2 In health economics, reclassication risk is often referred to as premium risk. See Arrow (1963), Pauly et al. (1995) and Cochrance (1995). See Diamond (1992) as well as Dowd and Feldman (1992). 1

3 many interesting for a study of one-sided commitment. The German market environment is unique in several aspects. One of these aspects is that Germany has a social health insurance (SHI) system as well as a private health insurance (PHI) system coexisting side by side. In the statutory SHI system, nonprot insurers (called sickness funds) collect premiums from their policyholders and pay health care providers according to negotiated agreements. Consumers who are not insured through these funds, mostly civil servants and the self-employed, usually have private insurance. Indeed, less than 0.2 percent of the German population has no health insurance of any kind. 3 These consumers are generally the very rich who do not need it, and the very poor who receive health care through social assistance. SHI premiums are not risk-based but depend on an individual's annual labor income. Children and spouses without labor income are insured without surcharge. Premiums are shared between the insured and his or her employer. Sickness funds are required by law to set a uniform premium for all their policyholders. The most important features of the German SHI system are community rating and open enrollment, i.e., to ensure coverage for all risk types, sickness funds are required to accept any individual who applies without making a risk assessment. 4 The PHI system is described below. Another unique aspect of the German PHI system is its high degree of government regulation. The remainder of the paper proceeds as follows. The next section reviews related literature. The basic structure of the German private health insurance system and its functioning is explained in Section 3. We derive our main hypotheses according to the theory of one-sided commitment in Section 4. Section 4 also contains the empirical analysis. Section 5 discusses policy implications and concludes. 2 Related Literature The paper is related to two strands of literature. The rst strand is the vast literature on dynamic contracts, learning and commitment. This theoretical literature has received relatively little empirical attention. Hendel and Lizzeri (2003), following Harris and Holmstrom (1982), test a theory of dynamic contracting in the U.S. life insurance market. They provide strong evidence of the existence and signicance of learning over time. Life insurance contracts involve front-loading (prepayment of premiums) resulting from a lack of bilateral commitment to contracts. This front-loading creates a partial lock-in for consumers and more front-loaded contracts generally involve lower lapsation. Finkelstein, McGarry and Su (2005) support Hendel and Lizzeri's ndings using data on the U.S. long-term care insurance market where mortality risks are learned over time. However, there are several important dierences between 3 4 See Thomson, Busse, and Mossialos (2002), p Government regulation is an alternative way to insure premium risk. When community rating is required, health insurers must impose a somewhat uniform price for all individuals who enroll in their health insurance plans. Community rating is then necessarily associated with (a) open enrollment and (b) compulsory insurance. Open enrollment is necessary to avoid cherry picking by insurers. Compulsory insurance is necessary to ensure cross-subsidization between high and low risks. This is because if insurance were not compulsory, low risks would prefer to purchase risk-based insurance (or remain uninsured) to avoid cross-subsidizing high risks. Government regulation in this form - a combination of community rating, open enrollment and compulsory insurance - is implemented in Belgium, the Netherlands, and Switzerland, and is part of Enthoven's (1988) proposal to reform the U.S. health care system. See Kifmann (2002), p. 15. Since our focus is on private health insurance in Germany, we do not further discuss Germany's SHI system. For a more detailed discussion, the reader is referred to Breyer (2004). 2

4 life insurance contracts and health insurance contracts. A life insurance contract mainly insures an income stream; a health insurance contract insures health care treatment. Learning about health is an important phenomenon. Yet life insurance contracts are comparatively simple and explicit when compared to health insurance contracts, and therefore asymmetric information is not an important issue. 5 When we look at more sophisticated health insurance contracts evolving over time, risk selection may be more important as distortions due to dynamic information revelation on health status can be large. Herring and Pauly (2006) extend the work of Hendel and Lizzeri (2003) and show that essentially the same results can be obtained with guaranteed renewability and single-year individual health insurance policies. They also provide comparisons of the extent of front-loading in premiums paid with estimates of the optimal age-path of premiums. It is shown that the extent of front-loading necessary to fund guaranteed renewable policies is reduced by several factors so that the incentive-compatible premium schedule increases with age. The second strand of literature this paper contributes to is the theory of adverse selection in insurance markets. 6 Following this theory, high-risk agents can be expected to purchase more (comprehensive) insurance coverage. This prediction, the coverage-risk correlation, can also be expected to manifest itself in a greater tendency of high-risk agents to purchase insurance. 7 A high-risk individual is one who generates higher expected insurance payouts due to a larger number of expected claims, a higher expected payout in the event of a claim, or both. Empirical evidence on adverse selection in insurance markets varies across markets and pools of policies. A signicant amount of empirical work nds evidence of adverse selection in insurance markets. 8 Early research by Phelps (1976) reports no evidence of a signicant relationship between predicted illness of individuals and their choice of insurance coverage. Although Cardon and Hendel (2001) suggest that informational asymmetry in the U.S. health insurance market may be unimportant, Browne (1992) provides statistical evidence that adverse selection is present in the market for individual health insurance in the U.S.. His analysis conrms that cross-subsidization of high risks by low risks occurs in this market. Browne and Doerpinghaus (1993) also focus on the U.S. market for individual health insurance. They nd that the characteristics of the insurance policies purchased by high and low risks and the premiums paid by high and low risks are similar, but that high risks derive more indemnity benets from their insurance contract than do low risks. This nding suggests that adverse selection in the market for individual health insurance results in a pooling of risk types. Browne and Doerpinghaus (1994) report similar results in a study of the Medicare supplemental insurance market in the United States. Cutler and Reber (1998) study dierent health insurance plans See Hendel and Lizzeri (2003), p The theory of adverse selection in insurance markets was introduced by Rothschild and Stiglitz (1976) and has been extended in many ways. For a survey and discussion of theoretical adverse selection models, see Dionne, Doherty and Fombaron (2001). The existence of a coverage-risk correlation itself is viewed as necessary for adverse selection to be present (and its absence as sucient for rejecting adverse selection). See, for instance, Chiappori and Salanié (2000). However, it may not always be sucient to conrm the existence of adverse selection given that this correlation is also suggested by moral hazard theory. See, for instance, the discussion in Cohen and Siegelman (2010), pp To test for adverse selection, individuals facing the same set of choices should be examined. To test for moral hazard, similar individuals facing dierent coinsurance rates should be examined, where price sensitivity can be measured by using the coinsurance variability across individuals. See, e.g., Cardon and Hendel (2001). A detailed discussion can be found in Cohen and Siegelman (2010). For a detailed review, see Cutler and Zeckhauser (2000). They review a substantial amount of empirical adverse selection studies in health insurance. Virtually all of these studies support the hypothesis of informational asymmetry in favor of policyholders. While our focus is on health, adverse selection is shown to be present in several other insurance markets. See, for instance, Makki and Somwaru (2001) or Cohen (2005). 3

5 oered by Harvard University, which switched from subsidizing the most generous plans to oering a xed-dollar subsidy, thus increasing the annual cost of the most generous plan. The coverage-risk correlation was strongly conrmed by their analysis: the most generous plan was abandoned by the best risks. Finkelstein and McGarry (2006) nd no statistically signicant evidence of a positive correlation between insurance coverage and (ex post) realizations of loss in the long-term care insurance market in the United States; however, Browne (2006) shows that high-risk types are more likely to retain their insurance coverage, an indication of the presence of adverse selection in this market. Although there is a signicant body of empirical work addressing adverse selection in health insurance markets, the main focus of this literature has been on testing predictions of static insurance models. 9 An exception is Dionne and Doherty (1994), who study a two-period competitive insurance market with one-sided commitment and renegotiation. They show that an optimal renegotiation-proof contract may entail semi-pooling in the rst, and separation of risk types in the second period. More importantly, they show that optimal contracts will exhibit highballing features, i.e., the insurer will typically make positive prots in the rst period, compensated by below-cost second period contracts. This study contributes to the smaller literature on dynamic contract theory, learning, and one-sided commitment. We empirically investigate contract dynamics in a world where consumers cannot commit to renewing a contract but insurers commit to oering renewal at a premium rate that does not reect revealed future information about the insured risk. 10 We combine the two strands of literature, that on one-sided commitment and learning with that on adverse selection. 3 Private Health Insurance in Germany 3.1 The PHI market environment About 10 per-cent of all Germans have private health insurance. Only an upper-income group as well as the self-employed and civil servants are eligible for the private system. 11 Employees can purchase PHI only if their annual labor income exceeds a certain threshold. 12 If annual income is below this social security ceiling, they must remain within the compulsory community-rated public health insurance system. There are exceptions for learners and students who can more easily choose between both systems due to the absence of income See the seminal theoretical work by Akerlof (1970), Rothschild and Stiglitz (1976) and Wilson (1977). To our knowledge, there is no empirical study on adverse selection in the private health insurance market in Germany to date, but Nuscheler and Knaus (2005) examine risk selection within the German social health insurance system. They look at why company-based sickness funds were able to attract many new customers during and study potential determinants of switching behavior. They nd no evidence for selection by sickness funds in German SHI. Self-employed and civil servants are not compulsorily insured in the SHI system. For the latter, there is an entirely tax-nanced plan for civil servants (called Beihilfe). Depending on marital status and the number of children, this plan covers % of health care expenditures with the remainder being covered by an additionally purchased PHI contract. As civil servants would lose these entitlements while staying in the public system, there is a strong incentive to (partially) join the PHI system. In 2010, for instance, annual income before taxes must exceed 49,950 Euros. 4

6 restrictions. 13 In case a policyholder retires, the income ceiling does no longer apply and she stays in the private system even though her income may have fallen below the ceiling. 14 The German PHI market is rather oligopolistic. The German Association of Private Health Insurers (PKV Verband) reported 45 members by the end of The legal form of private health insurers is either a stock company or a mutual. There were 26 stock companies and 19 mutuals insuring approximately 8.8 million people by the end of Overall premium income was million Euros for individual private health insurance coverage. Women have a lower share of private health insurance coverage than men. This is because women tend to have a lower annual labor income and thus do not cross the income threshold to be eligible for private health insurance as often as men do. 16 There is migration within and between these two health insurance systems. However, switching possibilities from the private to the social system and vice versa are somewhat restricted. Individuals aged 55 or older are not allowed to switch to the SHI system in any case. Despite regular increases in the contribution ceiling, ever since 1975, the number of people switching to substitutive private health insurance has been higher every year than the number of people lost to the statutory social system. Since 1997, however, the number of people switching from social to private health insurance has seen a substantial increase. The German Association of Private Health Insurers argues that this is partly due to cutbacks in the statutory SHI system. There are no data available on the extent of migration within the PHI system. However, ineciencies from the lack of bilateral commitment may involve migration-inducing distortions. In view of many countries' current health insurance reform debate, it is of interest to study and evaluate these potential distortions. 3.2 PHI premium calculation The private health insurance system in Germany is a system that oers comprehensive health insurance (including the cost of outpatient, hospital, and dental treatment) for the whole life. This is important since it means that an insurer, when calculating premiums, needs to take into account all dierent life periods of the individual who applies for coverage. In contrast to the SHI system where premiums depend on income, PHI premiums are risk-based, unrelated to income, and individually calculated using a funding principle. Policyholders accumulate funding capital to compensate for higher expected health expenditures in the future. From an actuarial viewpoint, a constant net premium is calculated such that accumulated aging provisions in early contract years are sucient to compensate high health care costs in later years. 17 Therefore, as health expenses increase with age, premiums necessarily exceed expected Most children are insured via their parents' PHI or SHI contract. However, they might change this status and switch systems or contracts when they enter the job market. For instance, it is possible to pause a PHI contract, when a 16-year-old decides to enter an apprenticeship or other work during which he will be insured in the SHI system. Therefore, there is a choice even for young people in choosing between systems or dierent PHI contracts. Note that once a policyholder has entered a PHI contract, he or she cannot simply drop out and reenter the SHI system, regardless of how attractive this seems. The policyholder must stay in the PHI system as long as annual income is above the ceiling. However, it is possible to switch within the system and to choose any other PHI contract oered by either the same or another private health insurer. Taking also into account 31 comparably very small non-members, the overall number of companies in the market amounts to 76. See German Association of Private Health Insurers (2009/2010), pp. 17,29. Insurers calculate with a maximum life expectancy of 102. Since life expectancy depends on age and risk type of a policyholder, premiums vary for dierent ages at entry. 5

7 cost in early years and fall below expected cost in older years. German law requires, however, that private health insurers calculate premiums in such a way that they are constant over the insured's life-time. 18 The precautionary savings element used to smooth premiums over time is called the aging provision or aging reserve. Since premiums are risk-based at contract entry, some risk assessment is needed. The insurer conducts such a risk assessment at initial enrollment. The resulting premium then depends on overall health status, sex, age at entry as well as on the extent of PHI coverage chosen. In particular, when a policyholder enters into a new PHI contract, the risk assessment can lead the insurer to imposing some risk loading on the net premium due to the individual's poor health status. Health impairments are usually assessed by a health questionnaire and/or doctor's report. However, there is no reassessment of risk type over time and the premium loading generally stays constant. 19 As a consequence, PHI premiums are not adjusted over time according to risk type and policyholders face no reclassication risk. Indeed, they only face this risk if they decide to switch their insurer. Then, again, the new insurer conducts an individual risk assessment and may impose some risk loading in the premium due to poor health at contract entry. When a policy lapses (when the insured dies or switches insurers) the aging provisions accumulated up to that point is forfeited in favor of the remaining insured community. So if policyholders wish to switch insurers, they cannot take any fraction of the aging provision with them. 20 It is important to note that the aging provision is dened as a collective reserve and does not belong to an individual policyholder. The insured is thus not entitled to the surrender value. As a result, there is implicit partial cross-subsidization between policies over time and some benet of survivorship. However, since there is always an individual risk assessment at contract entry with a new insurer, switching is more attractive for lower-risk types. The funding principle used to calculate PHI premiums is implemented via a basic actuarial rule, the so-called principle of equivalence. This principle states that over the entire policy duration (generally life-long) the total of the premiums must match the total of the benets, including expenses caused by writing and administration of the policy, for each category of equivalent risk. The magnitude of the individual health risk is determined by the benets under the contract, the policyholder's age (at contract entry) and the policyholder's sex. Theoretically, the premium remains constant throughout the policyholder's life time as long as the actual benets match those used to calculate the premium. In reality, cost increases in health care cause the benets and thus the premiums to change throughout the insured period. As a result, the principle of equivalence is static and only fullled at the moment of calculation. 21 More formally, the principle of equivalence states that a lifetime constant net premium is calculated such that the present value of expected premium income (P ) is equivalent to the present value (P V ) of expected calculated health expenditures (CHE). It is shown in the equivalence equation: E[P V (P )] = E[P V (CHE)] (1) In practice, premiums are not constant but depend on external factors. Premiums would be constant if, for instance, the insurer's insured community and treatment cost did not change. See Milbrodt (2005). If the policyholder can prove that the precondition(s) that led the insurer to impose a risk loading is no longer present or has become unimportant, the risk loading can be reduced. It should be noted that this fact has partly changed since Since January 2009 private health insurers oer an additional base rate (Basistarif). Under certain circumstances, switching into such a contract is possible without losing accumulated aging provisions. Since our empirical study is based on data that were collected before 2009, this new law has no impact on our study and we thus neglect it. See Fuerhaupter and Brechtmann (2002). 6

8 expenditures (ECHE). It is shown in the equivalence equation PV [EP] =PV[ECHE] , ,00 Amount in Euro 12000, , , , ,00 Niedrige Low health Kopfschadenreihe expenditures per risk Hohe High Kopfschadenreihe health expenditures per risk Premium for age at entry 30 and Prämie bei Eintrittsalter 30 und low health expenditures per risk niedriger Kopfschadenreihe Prämie Premium bei Eintrittsalter for age at entry 30 und 30 and hoher high health Kopfschadenreihe expenditures per risk Prämie Premium bei Eintrittsalter for age at entry 50 und 50 and niedriger low health Kopfschadenreihe expenditures per risk Prämie Premium bei Eintrittsalter for age at entry 50 und 50 and hoher high health Kopfschadenreihe expenditures per risk 2000, , Age Figure 1: Health expenditures per risk and net premiums. Source: Rosenbrock (2010). 100 In this equivalence equation, the present values are determined by making use of - mortality tables, - an actuarial interest rate, - health expenditures (claims per capita), - other patterns of lapses except death. Following legislation, the technical interest rate must not exceed 3.5%. Using some exemplied progression data, Figure 1 illustrates how premiums are determined. 22 As can be seen from the gure, higher health expenditures per risk imply higher premiums over the insured's lifetime. Premiums are based on the age at entry of the policyholder. Of course, for a higher age at entry, premiums are increased since there is less time remaining for pre-nancing health expenditures in the future. Remember that parts of the premium not used for indemnity payments due to lower health expenditures per risk in younger years are accumulated by the insurer in the form of actuarial aging provisions. Due to interest eects, aging provisions tend to increase until policyholders reach a high age (see Figure 2) even though the health expenditures per risk curve crosses the premium curve signicantly before this time (see Figure 1). The accumulation of aging provisions ensures that at each point in time the equivalence equation holds, i.e., the present value of expected premium income and existing aging provisions (AP) is equivalent to the present value of expected calculated health expenditures E[P V (P )] = E[P V (CHE)] AP. (2) which constitutes the generalized equivalence equation. This equation is the actuarial basis of individual premium calculation. 23 AP depends only on age, sex, policy duration, and extent of coverage chosen. Individual reclassication risk is insured because equation (2) The gure stems from Nell and Rosenbrock (2009). The calculations in Figure 1 are notional and follow Milbrodt (2005). Calculations include exit and mortality risk as well as an actuarial interest rate. Note that for a new policyholder, equation (2) corresponds to equation (1) with AP = 0. Thus, equation (1) is a special case of equation (2). 7

9 Figure 2: Aging provisions for dierent ages of entry and health expenditures per risk. Source: Rosenbrock (2010). does not include any individual health information: it follows directly from the actuarial calculation principle. Thus, when the policyholder's state of health deteriorates, the premium stays constant. The basic mechanism is as follows. After a few periods (theoretically, after each period), CHE and mortality rates used for calculating present values generally dier and need to be recalculated. Then, if necessary, aging provisions are adjusted. 24 Together with new mortality tables, the new premium P is calculated and a new equivalence equation as shown in equation (2) is obtained. Individual reclassication risk is insured because CHE and mortality tables apply to all policyholders in the insurer's collective, and adjustment of aging provisions is regulated so that individual premiums do not vary according to individual health status. In contrast to individual reclassication risk, we use the term collective reclassication risk to mean the risk that collective health expenditures exceed collective costs. This risk is borne by policyholders via premium loadings since it follows the development of health expenditures in the insurer's collective. Premiums need to be adjusted if overall premium income does not suce to cover overall health expenditures. While individual reclassication risk can be insured via partial risk pooling, i.e., using a collective actuarial calculation method, collective reclassication risk cannot be insured. In a long-term perspective, dynamic information revelation as to health status informs policyholders about their individual risk. Since the equivalence principle is based on collective actuarial calculation, and aging provisions do not include any information on individual health status, the system appears prone to risk selection. This is because aging provisions are objectively too high for low risks and too low for high risks. It seems likely that lower-risk types, once they discover that they are low risks, will be inclined to cancel their policy and look for cheaper and less comprehensive coverage elsewhere. Therefore, reclassication risk constitutes some implicit switching cost and partial risk pooling seems likely to entail risk selection in the German PHI market. 24 Methods for adjusting aging provisions are regulated. Financial resources for this adjustment stem from separate sources or insurers' nancial surplus. 8

10 There are ve possible reasons a policyholder may opt out of a PHI contract. Individuals opt out over time (a) because their incomes have fallen below the contribution ceiling, (b) because they nd out they are low risk and switch to another more favorable health insurance contract involving less extensive coverage and lower prices (c) because their work status changes from self-employed to wage earner, 25, (d) because they expatriate themselves, or (e) because they die. Of course, we cannot dierentiate between the causes for lapsation, but we may draw conclusions about relationships. In this view, our focus will mainly be on the (b), the second reason. 4 Model Framework and Empirical Analysis German PHI premiums involve front-loading (prepayment of premiums). As a consequence, policyholders transfer income from early contract years to later years, during which they generally have worse health. Contracts are unilateral in the sense that policyholders can cancel their PHI policies, whereas insurers commit to the terms of the contract as long as the contract is in force. These contract characteristics are consistent with a dynamic model of one-sided commitment. To make theoretical predictions, it seems fruitful to adapt such a model framework, which we accomplish by relying heavily on Hendel and Lizzeri (2003). The key features of the model are (1) symmetric learning, i.e., information about risk type is revealed over time; (2) one-sided commitment of insurance companies; (3) buyer heterogeneity as to front-loading, i.e., consumers vary in income (growth); and (4) guarantee of full insurance against reclassication risk, i.e., premiums are independent of health status. The following predictions about the competitive market equilibrium set of contracts, which can be found in the Appendix, are an adaptation of Hendel and Lizzeri (2003) to our environment. All consumers obtain full insurance in all possible states of the world. In the second period, premiums involve a cap so that the actual premium is below the fair premium in this period. Consumers transfer income from the rst period, when they enjoy comparatively good health, to future states involving worse health. The initial overpayment in the premium creates a lock-in or commitment to the PHI contract. This renders switching to a rival insurer unattractive for policyholders. Contracts are front-loaded as long as income growth is below a certain threshold. Less front-loaded contracts appeal to buyers with lower rst-period income. It should be noted that these predictions are drawn from equilibrium allocations via fully contingent contracts that involve no lapsation. However, there are comparable non-contingent contracts that allows us to derive comparative statics predictions. Non-contingent contracts with higher rst-period premiums (i.e., higher front-loading) are chosen by consumers with lower income growth and have a lower rate of lapsation. 25 In this case, individuals can either cancel or pause their PHI contract. Pausing may be attractive if the work status may change again in the future. Then, the individual can reenter the PHI contract at the same conditions. If, however, the person feels that he or she is a low risk and that the SHI system can do equally well, the person may cancel the PHI contract. We take this eect into account in our empirical analysis below. 9

11 The PHI market we study is dierent from the life insurance market studied by Hendel and Lizzeri (2003). However, the basic mechanism behind the market forces is very similar. German PHI contracts insure individual reclassication risk. A policyholder faces reclassication risk only when he or she considers entering a PHI contract with a dierent insurer. In summary, we predict the following two key hypotheses for the German PHI market Front-loading creates a lock-in of consumers, i.e. contracts with higher front-loading have lower rates of lapsation due to a more severe lock-in, other things equal. 2. Since high-risk types have a lower incentive to lapse for any given contract, ceteris paribus, the risk pool worsens over time. An important characteristic of German PHI premiums is that they are at. This coincides with the most front-loaded life insurance contracts observed in the US and Canada. Interestingly, another implication of the model is that the most front-loaded contracts are at. 4.1 Data We study enrollment in a comprehensive private health insurance contract over a period of ve years, 2001 through 2005, using a large sample data set from a German private health insurer. To analyze potential ineciencies resulting from the lack of bilateral commitment in this market, it is of interest to discover which individuals drop their PHI contract over time. Therefore, we create a subsample consisting of those 5, 681 individuals who were enrolled with the insurer in 2001, and then study the characteristics of those individuals over the following ve sample periods. The subsample used for our statistical analysis contains 28, 405 (i.e., 5, 681 5) consumer-year observations. The sample includes information on gender, age, tenure, pausation, individual health expenditures, and insurance premiums paid, but information on other characteristics of interest, such as marital status of a policyholder, race, and income, is not collected by the insurer. A short overview of the sample is shown in Table Total enrollment 5,681 4,871 4,256 3,859 3,627 Enrollment male 4,143 3,558 3,110 2,830 2,651 Enrollment female 1,538 1,313 1,146 1, Average premium 1, , , , , Average premium male 1, , , , , Average premium female 2, , , , , Average loss 1, , , , , Average loss male 1, , , , , Average loss female 1, , , , , Table 1: Data overview Women are higher risk type than men on average. In terms of loss frequency, over the ve periods observed, per-cent of women made health insurance claims, whereas only per-cent of men did so. Women are also higher risk type in terms of loss severity. The average 26 Note that we observe a given number of policyholders over time. In practice, new policyholders joining the collective tend to dilute eects. 10

12 annual treatment cost for a woman in our sample is 2, Euros; for men, the average health care cost is 1, Euros. As a consequence of their being a higher risk type, women pay, on average, about 20 per-cent more for their health insurance premium than do men. Histogram of Birthyear Frequency Birthyear Figure 3: Year of birth of policyholders. Figure 3 displays the distribution of birth year for the policyholders in our sample. Most policyholders were born between 1965 and 1970, meaning that most were in their early thirties in the study period The average age of a policyholder in our sample is Since the regulatory framework of German PHI requires employed individuals to stay within the social health insurance system as long as their earnings do not exceed the PHI income threshold, the majority of PHI participants are around 30 years old. 27 However, given that newborn children tend to enter via their parents' PHI contract, this average is lower when children are taken into account. We include children in our study (this is useful for predicting claims in the prediction model, and results of the test model do not change without children). The average age of entry is We are interested in the characteristics of policyholders who cancel their policy. We conduct our analysis in two steps. The rst step involves using an estimation model to obtain a proxy for risk type in our sample. We refer to this model as the Prediction Model. We obtain this proxy before we test our theoretical predictions, which is the second part of our analysis. We refer to the second model as the Test Model. In the rst-step modeling approach, we use a two-part model to predict expected medical expenditure, which is our proxy for a policyholder's risk type. In the second step, we use logistic regression to test our hypotheses. Each statistical method is explained in more detail below. 4.2 Predicting Risk Type via Expected Medical Expenses To test our hypotheses, it is necessary to classify all policyholders in the sample by their risk type. Although individual risk type is unobservable, claims, along with their frequency and severity, can be observed. We will use (predicted) medical expenses as a proxy for risk type. Having identied risk types, we then look at which policyholders drop their coverage using this risk type proxy. 27 See Baumann et al. (2006), p

13 Since the distribution of actual expenses has a large mass at zero and is heavily skewed, we estimate individual-level medical expenses using a two-part regression model for health expenditures. Compared with ordinary regression, which ignores the special pattern of a large share of zeros in the dependent variable, a two-part model can provide an unbiased estimation. In a two-part model, the frequency component and severity component are modeled separately. Following the traditional actuarial literature based on the individual risk model, the response, i.e., the insurance claim, can be decomposed into two components: a frequency (number) component and a severity (amount) component. 28 More formally, let r i be a binary variable indicating whether the ith individual had an insurance claim and let y i describe the amount of the claim given there was a claim. Indeed, the mechanism that determines zero or nonzero expenditures might not be the same as the mechanism that determines the amount of positive expenditures. Then, the claim can be modeled as (claim recorded) i = r i y i (3) constituting the two-part or frequency-severity model. 29 The rst part is a logit model predicting for the likelihood of having any nonzero medical expenses, the second part is a linear regression for the logarithm of annual medical expenses for the sub-sample with nonzero expenses. 30 Our aim is to predict logarithmic medical expenses which is our proxy for policyholder risk type. 31 Specically, our rst equation is a logit equation for the dichotomous event of zero versus positive annual medical expenditure. Following Manning et al. (1987), we express the expectation of the response as a function of explanatory variables to be the probability of a claim, i.e., P rob(claim i = 1) = exp(x i β) 1 + exp(x i β) (4) where Claim i is a binary variable equal to one if there is a claim for a given insured individual i; zero otherwise. The unknown parameters β j are estimated by using maximum likelihood techniques. Similarly, our second equation is a linear regression on the log scale for positive medical expenditure given that the policyholder receives any medical services: Lnloss i Loss i > 0 = x iδ + ɛ i. (5) Assuming that β and δ are not related, i.e., under independence, the two parts of our model are estimated separately to result in a prediction of expected medical expenditures per policyholder in a given period. The list of variables we use for our analysis is in Table 2, and the resulting estimates for loss frequency and severity, respectively, are shown in Table See Bowers et al. (1997), ch. 2. See Frees (2010), p The logarithmic transformation nearly eliminates the typical undesirable skewness in the distribution of medical expenditures, making the model more robust. In particular, it yields nearly symmetric and roughly normal error distributions, for which the least squares estimate is ecient. See Duan et al. (1983). Note that we could also predict medical expenditures (without the log scale) but this would require a retransformation to the normal scale. A shortcoming of this procedure is that the error terms in the log expenditures equation are often not normally distributed but still skewed so that normal retransformation estimates are biased. Therefore, the smearing estimate, developed by Duan (1983), is often used to estimate the retransformation factor. The smearing estimate is given by the sample average of the exponentiated least squares residuals. Yet estimating expected expenditures in this way complicates our analysis without providing further insights: since there is a positive relationship between expected losses and expected logarithmic losses (and since we only need a risk classication proxy here) we can use predicted expenses on the log scale as a proxy for risk type. 12

14 Variables in the Prediction Model Prediction Variables Control Variables Response Variable Control Variables Type Min. Max. Mean Std.Dev. Claim (Claim = 1) Binary Lnloss Numerical Female (Female = 1) Dummy Age Numerical Agesquare (Age*Age) Numerical CHE Numerical CHE_Fem (CHE*Female) Numerical Contractyear Numerical Riskload Numerical Exposure Numerical Year Numerical Variables in the Test Model Enrol (Enrol = 1) Binary Female (Female = 1) Dummy Agecat0 (age category 0 = 1) dropped Agecat1 (age category 1 = 1) (omitted) Agecat2 (age category 2 = 1) Dummy Agecat3 (age category 3 = 1) Dummy Agecat4 (age category 4 = 1) Dummy Agecat5 (age category 5 = 1) Dummy Shortdur Dummy Middur (omitted) Longdur Dummy Entryage Numerical Medexp Numerical Table 2: Response and explanatory variables in the empirical models. Claim is a dichotomous variable equal to one if there has been a claim in period t; zero otherwise. We dene a claim as being observed when medical expenditures are nonzero. Lnloss is a numerical variable indicating the size of medical expenditures on the log scale for a given period. It is dened only in the event of a claim, i.e., when medical expenditures are nonzero. Female is a dichotomous variable equal to one if the policyholder if female; zero otherwise. Age indicates the age in years of a policyholder in a given period. Similarly, Agesquare is the square of Age. CHE was explained in section 3.2 above. It is a numerical variable indicating calculated health expenditures per insured for a given tari in a given year. In other words, CHE is what the insurer expects to spend in t on a given policyholder taking into account the policyholder's age and sex. 32 CHE_Fem represents an interactions term. Contractyear is a numerical variable indicating the policy duration. For instance, at contract entry, Contractyear is equal to one, meaning that the contract is in its rst duration period. Year is a count variable indicating the period, i.e., 2001 through Riskload is a numerical variable representing the premium loading for a high-risk policy, that is, Riskload is positive when the insurer conducts a risk assessment at contract entry and nds that the policyholder 32 According to Ÿ6 of the Order of Premium Calculation Methods in Private Health Insurance in Germany (Kalkulationsverordnung - KalV), CHE is dened as average benets per insured. It needs to be calculated based on age and sex of the policyholder for a given period (a year) and tari. The calculation of CHE must take into account former claims and actuarial methods must be used in order to smooth random uctuation. 13

15 is actually high risk. The size of Riskload depends on the specic precondition or illness the policyholder suers. Exposure is a numerical variable indicating the number of months a PHI contract was actually in force for a given period. This variable takes into account that (1) a contract may be paused for some time and thus there is no insurance coverage in case of a loss, and (2) an individual may have joined the pool later in the year. Enrol is a dichotomous variable equal to one if a policyholder is enrolled with the insurer in a given period; zero otherwise. Entryage is a numerical variable indicating the age at entry of the policyholder. Shortdur, Middur, and Longdur are Dummys dividing policyholders into three groups: those with short contract duration of 0-14 years, those with middle contract duraction of years, and nally those with long contract duration of over 30 years. Table 3: Loss frequency (1) and severity (2) estimations. We group individuals according to their age. Age category 0 includes children (individuals with age below 18). Category 1 includes young adults of age Age category 2 includes individuals aged 25 to 34. Age category 3 includes individuals aged 35 to 44, category 4 includes ages Finally, age category 5 includes all individuals older than According to these categories, Agecat1 is a dummy variable equal to one if the age of a policyholder falls within age category 1 for a given period; zero otherwise. Agecat2 through Agecat5 are similarly dened. The two-part model estimations conrm that women are altogether higher risk than men. The coecient on loss probability (Claim) is positive and signicant at the 5 per-cent level. 33 Note that this nal age category makes sense when taking into account that, at some age above 50, switching is no longer possible (following German law). 14

16 The coecient on loss severity (Lnloss) is positive but not signicant. Higher age tends to have a positive (but decreasing) eect on loss severity and a negative (but increasing) eect on loss probability. This suggests that treatment cost increases with age but policyholders tend to visit physicians less often. Our estimations conrm the intuition that the insurer's expected calculated health expenditures are positively correlated with both loss probability and loss severity. A higher exposure during a given period, that is, a longer policy duration, is associated with higher loss probability and severity. Finally, a higher risk loading due to bad health status has a positive eect on both loss probability and loss severity. As also clear from the estimated parameters of the twopart model for loss severity prediction, the R 2 does not explain very much of the variation in loss probability and severity. However, this is not surprising as insured losses are largely random events and thus a considerable amount of unexplained variation is expected. 34 Using these estimates and combining them in a new single estimate, Medexp, which is simply the product of the frequency and the severity components, we obtain an estimate for expected medical expenditures (on the log scale) for each policyholder. We will use this estimate as an independent variable in the second part of our analysis when we test our hypotheses regarding the German PHI market. 4.3 Testing the Implications of the Model We consider the individual's decision to enroll in a PHI contract as a discrete choice. An individual may either stay with the original contract or enter a new PHI contract with some other (private or public) insurer. 35 The net expected utility of being enrolled in the PHI contract as compared to switching to some other contract is assumed to be given by some linear index function EU i = x iφ + ɛ i, (6) where x i denotes individual i's characteristics, φ is the vector of parameters to be estimated and represents the impact of these characteristics on the decision to stay enrolled in the private health insurance contract, and ɛ i is a random error term. We do not observe the net expected utility of being enrolled in the private health insurance contract in a given period, but we do observe whether the net expected utility is positive, meaning that the individual decided to be enrolled in the contract in a given period. Thus, using logistic regression analysis, the probability that an individual is enrolled in a given period t is modeled as: P rob(enrol i,t = 1) = P rob(eu i,t > 0) = exp(x i φ) 1 + exp(x (7) iφ), where Enrol i,t is a dummy variable equal to one if individual i is insured in period t; zero otherwise. The logit modeling approach here consists in estimating the probability that a policyholder drops out of his or her PHI contract in period t, depending on individual characteristics. We use predicted medical expenditure (on the log scale), M edexp, as proxy for a policyholder's risk type. Logistic regression results for the test equation are shown in Table 4. The test equation is estimated four times for each data period in the sample based on 100 per-cent enrollment Note that we are not using the prediction equation to test any hypotheses. Therefore, collinearity is not a concern at this point of our analysis. Note that this is true for over 99.8 per cent of all Germans since the proportion of uninsured people in Germany is below 0.2 per cent. Therefore, we ignore the decision to drop health insurance coverage. 15

17 in the previous period, i.e., we evaluate lapsation for every period from 2002 through Therefore, in our test equation below, the number of observations available for testing in period t will correspond to enrollment in t-1. Age category 1, i.e., young policyholders aged 18 24, is the reference age category and left out of regression. Interpretations will thus refer to age category 1 as the reference age group. 36 Mean variance ination factors range from 2.25 to 3.39 for all test periods. (2002) (2003) (2004) (2005) VARIABLES Enrol02 Enrol03 Enrol04 Enrol05 Female **** **** **** * (0.294) (0.274) (0.301) (0.325) Agecat **** 3.913**** 4.002**** 3.288**** (0.607) (0.534) (0.646) (0.632) Agecat **** 5.796**** 5.752**** 4.243**** (0.833) (0.693) (0.816) (0.768) Agecat **** 7.556**** 7.490**** 5.433**** (1.007) (0.853) (1.004) (0.910) Agecat **** 8.121**** 7.630**** 4.454**** (1.501) (1.032) (1.145) (1.228) Entryage ** * (0.0133) (0.0113) (0.0124) (0.0128) Medexp 10.57**** 7.599**** 7.094**** 4.801**** (0.923) (0.747) (0.934) (0.797) Constant **** **** **** **** (1.518) (1.218) (1.452) (1.259) Observations Pseudo R 2 4, , , Standard errors in parentheses **** p<0.001, *** p<0.01, ** p<0.05, * p<0.1 3, Table 4: Enrollment Mean choice Vifs range estimation from 2.87 results to 4.11 (test equation). We now address the hypotheses derived above. Theoretical predictions concerning one-sided commitment in a market environment such as German PHI insurance suggest the following. (1) Front-loading creates a lock-in of consumers: contracts with higher front-loading suer lower rates of lapsation due to a more severe lock-in. (2) Low-risk policyholders have a higher incentive to lapse. Since low-risk types have higher lapsation, the risk pool worsens over time. In regard to the rst hypothesis, we expect that more front-loaded contracts will experience lower rates of lapsation. Since we do not have information about prepayments (aging provisions) for policyholders, we need a proxy for front-loading. We use age at entry, i.e., the variable Entryage for this purpose. This is reasonable because for a given risk type, a higher entry age is associated with a higher premium. The premium is higher due to the fact that the insurer needs higher annual prepayments to account for increased health care cost in old age (see Figure 1). As a consequence, a higher entry age is associated with a more severe 36 Since we look at switching behavior of policyholders, it seems useful to consider only adults for the analysis, i.e., policyholders with a minimum age of 18. Then the number of observations would dier from Table 1. Instead, the reference category are children and young adults. Indeed, what matters is the parents' decision at the moment of choosing a health insurance plan. Thus children can be dropped in the test model without changing results because the focus is on switching behavior. Results are very similar with or without children. For convenience, we present the results including children here. 16

18 lock-in. Therefore, we would expect lapsation to be lower for higher entry ages in our sample. As can be seen from the test model results above, our data conrm this relationship, i.e., the coecient of Entryage is positive and statistically signicant in 2002 and 2003 indicating a higher tendency of a given risk type to retain coverage when he or she enters the contract in older age. The second hypothesis is that low-risk policyholders have a higher incentive to lapse. Our results conrm this hypothesis. The coecient of M edexp is positive and highly signicant over all tested periods. As a result, high-risk policyholders those with higher predicted medical expenditure M edexp are more likely to retain their PHI contract (which implies that low-risk types have a higher tendency to lapse). As a consequence, the risk pool worsens over time. In the theoretical section of this paper, we assumed perfect information. The insurer can observe individual health status via loss frequency and loss severity. Even if there was no important informational asymmetry between the parties, partial risk pooling and a lack of consumer commitment would distort ecient behavior. The positive and highly signicant coecient of Medexp implies that high risks are more likely to retain their PHI contract than are low-risk types. Such a worsening of the collective over time is indicative of adverse selection. Finally, it is interesting that the impact of gender in the form of F emale is negative. There are two possible explanations for this nding. Women might be higher risk since the cost of pregnancy is covered by the contract and included in the premium. This would be reected in higher premium dierences between men and women for age in the range of mid twenties to late thirties. Indeed, in our sample, age categories 2 and 3 exhibit a stronger average premium dierence between genders. Women have a signicantly lower tendency to stay with the insurer when they are aged 20 to 35. Another probable explanation is that the coecient represents an income eect. Since women tend to have lower income than men on average, and, as shown by our two-part model estimations, are also a higher risk type, their premiums are higher. Note that in our sample, women do indeed pay higher premiums (see Table 1). Interpreting F emale as a proxy for income, we would generally expect females to be more likely to cancel their PHI contract because their incomes are more likely to have fallen below the PHI contribution ceiling. This may be why their probability to retain a PHI contract is less than that of males. Second, if a woman drops out of employment to stay at home, she generally will be insured via her husband's insurance contract. This might have an impact on females' tendency to retain, or not, PHI coverage, too. One of the theoretical predictions we made earlier is that in a dynamic one-sided commitment market where contracts are not contingent on future health state, the equilibrium will be such that the contract with higher rst-period premium is chosen by consumers with lower income growth, and that this contract has lower lapsation. Given the structure of the German PHI market, premiums tend to increase with age of entry. Given that income growth is generally higher in age categories 1 and 2 compared to age categories 3 and 4, we would expect that age categories 3 and 4 (with lower income growth) will be required to pay higher prices (compared to age categories 1 and 2), and will exhibit a higher tendency to retain the high-priced contract (implying lower lapsation). This is exactly what we observe in the data: the coecients of age categories are positive and highly signicant, and they tend to increase from age category 2 up to age category 5 (see Table 4). Finally, we also used an alternative model to test enrollment choice in German phi. Naturally, 17

19 there is a high correlation between Contractyear and Entryage, i.e., a low age at contract entry is associated with a long contract duration in a given period. To avoid this multicollinearity problem, we divide policyholders into three groups: Shortdur, Middur, and Longdur are Dummys dividing policyholders into those with short contract duration of 0-14 years, those with middle contract duraction of years, and nally those with long contract duration of over 30 years. Middur is the reference category. The new regression results are shown in Table 5. Mean variance ination factors range from 4.83 to 5.48 for all test periods. Using Shortdur as a proxy for frontloading, the results show that policyholders with high frontloading, i.e., short duration, have a signicantly lower probability to keep their phi contract than the reference group. Thus there is higher lapsation in this group. Also, we nd that the group with long contract duration and thus relatively low (if not negative) frontloading has a signicantly higher probability to stay with their phi insurer. These results conrm our hypotheses. (2002) (2003) (2004) (2005) VARIABLES Enrol02 Enrol03 Enrol04 Enrol05 Female **** **** **** * (0.325) (0.295) (0.241) (0.293) Agecat **** 3.154**** 3.089**** 4.061**** (0.691) (0.576) (0.714) (0.774) Agecat **** 4.287**** 4.402**** 5.479**** (0.948) (0.785) (0.851) (0.948) Agecat **** 4.784**** 5.065**** 6.063**** (1.177) (1.024) (0.999) (1.109) Agecat *** 4.144*** 4.229**** 5.231**** (1.729) (1.335) (1.223) (1.286) Shortdur **** **** **** *** (0.848) (0.657) (0.531) (0.518) Longdur 2.196**** 1.594*** (0.659) (0.564) (0.621) (0.627) Entryage 0.266**** 0.164**** **** ** (0.0415) (0.0324) (0.0227) (0.0250) Medexp 13.62**** 8.816**** 3.595**** 4.162**** (1.255) (0.891) (0.391) (0.540) Constant **** **** **** **** (1.987) (1.410) (1.091) (1.315) Observations Pseudo R 2 4, , , Standard errors in parentheses **** p<0.001, *** p<0.01, ** p<0.05, * p<0.1 3, Table 5: Alternative enrollment choice estimation results (test equation). 5 Conclusion This paper addresses the issue of long-term insurance contracts in a world where risk types evolve over time, an especially important feature of the health and life insurance industries. The rst-best insurance policy in this world would fully insure against both short-term sickness risk and long-term reclassication risk (premium risk). Theoretical models of guaranteed 18

20 renewable insurance display front-loaded premium schedules. The paper contributes to the literature on one-sided commitment by studying properties of long-term private health insurance contracts in Germany. Private health insurance in Germany is regulated in such a way that insurers must oer long-term contracts at a guaranteed renewable rate involving front-loading of premiums and insurance of premium risk. The insurer accumulates aging provisions to smooth premiums over time. If policyholders want to switch insurers, they cannot take any fraction of this accumulated capital stock with them. As a result, this form of regulation creates a lock-in eect implying that switching is unattractive for policyholders who would bear a nancial loss even at early contract stages. As predicted by the theory on symmetric learning and dynamic contracting (Hendel and Lizzeri (2003)), our private health insurance data conrm that front-loading generates a lock-in of consumers, and more front-loading is generally associated with lower lapsation. Our results shed light on the nature of possible dynamic ineciencies in markets under one-sided commitment. We provide evidence suggesting that low-risk policyholders are more likely to drop their coverage, and that dropping coverage is at least partly a response to learning over time (i.e., low-risk policyholders discover positive information about their health status over time). Our analysis also supports evidence found by Finkelstein, McGarry and Su (2005) who studied long-term care insurance in the United States, a market that also involves learning (about mortality risk) over time. While symmetric learning has no eect on informational asymmetries between policyholders and insurers, some information may be private to policyholders who then may be able to more accurately estimate the severity and future consequences of their impaired health. Hence, in a world where reclassication risk is insured, given a specic mechanism, risk selection may be an issue. 37 Indeed, insuring reclassication risk may introduce a new problem: the stability of the pool must be ensured by new (better risk type) policyholders constantly entering the collective. This is due to the lack of bilateral commitment. Therefore, given imperfect commitment it seems necessary to collect enough funds upfront so as to enhance consumer commitment and minimize selection problems in the long run. The current study has several limitations that may be overcome by future analysis. First, the measure M edexp may not properly identify a policyholder's risk type in a relevant way. The relevant health conditions for a policyholder's renewal include those discovered after enrollment, but these are not included in the expense model. Of course, we do not have this information, but it seems critical for lapsing behavior. Second, our empirical model cannot include a measure of income, which is important as a general control and for the interaction with public health insurers. However, the insurer does not collect this sensible information from its policyholders, and the only information we have is that all policyholders in our empirical test model dispose of annual income which is above or equals the phi income threshold in Germany. Finally, another limitation of our study is that the data from this one private health insurer may not be representative of other German insurers. While the solutions to the premium risk problem proposed by Cochrane (1995) as well as Pauly et al. (1995) produce an incentive-compatible premium schedule preferable to facing 37 Life insurance contracts are comparatively simple and explicit; health insurance is more sophisticated, and thus asymmetric information seems a more important issue. In contrast to Hendel and Lizzeri (2003), we look at these more sophisticated contracts evolving over time. In this context, asymmetric information seems an important problem since distortions due to dynamic information revelation on health status can be large. Compared to Hendel and Lizzeri, who found that asymmetric information is not important in the case of life insurance, our results suggest that dynamic information revelation in health insurance may involve signicant risk selection. This may be particularly the case when premiums include partial risk pooling as in the case of Germany. 19

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